An equidistributed sequence is a sequence of real numbers 
on an interval ![[a,b]](/images/equations/EquidistributedSequence/Inline2.svg)
with 
for which the fraction of the first 
terms lying in any subinterval 
tends to 
as 
, where 
. The points of
an equidistributed sequence form a dense set on the
interval ![[a,b]](/images/equations/EquidistributedSequence/Inline9.svg)
.
However, dense sets need not necessarily be equidistributed. For example, 
,
where 
is the fractional part, is dense
in ![[0,1]](/images/equations/EquidistributedSequence/Inline12.svg)
but not equidistributed, as illustrated above for 
to 5000 (left) and 
to 
(right).
Hardy and Littlewood (1914) proved that the sequence 
,
of power fractional parts is equidistributed
for almost all real numbers 
(i.e., the exceptional set has
Lebesgue measure
zero). Exceptional numbers include the positive
integers, the silver ratio 
(Finch 2003), and the golden
ratio 
.
The top set of above plots show the values of 
for 
equal to e, the Euler-Mascheroni
constant 
,
the golden ratio 
, and pi. Similarly, the bottom set
of above plots show a histogram of the distribution
of 
for these constants. Note that while most settle down
to a uniform-appearing distribution, 
curiously appears nonuniform after 
iterations. Steinhaus (1999)
remarks that the highly uniform distribution of 
has its roots in the form of the continued
fraction for 
.
Now consider the number of empty intervals in the distribution of 
among the 
intervals bounded by 0, 
, 
, ..., 
, 1 for 
, 2, ..., summarized below for the constants
previously considered.
 | Sloane | # empty intervals
for  ,
2, ... |
 | A036412 | 0, 0, 0, 0, 1, 0, 0, 1, 1, 3, 1, 4, 4, 7, 5, ... |
 | A046157 | 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 3, 0, 3, 5, 3, ... |
 | A036414 | 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, ... |
 | A036416 | 0, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 4, 5, 7, 7, ... |
The values of 
for which no bins are left blank are given in the following table.
 | Sloane |  with no empty intervals |
 | A036413 | 1, 2, 3, 4, 6, 7, 32, 35, 39, 71, 465, 536, 1001, ... |
 | A046158 | 1, 2, 3, 5, 6, 7, 12, 19, 26, 97, 123, 149, 272, 395, ... |
 | A036415 | 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, ... |
 | A036417 | 1, 6, 7, 106, 112, 113, 33102, 33215, ... |
Kakutani partitions provide a related construction in which one sample point in the interior of each interval gives a sequence of finite point sets with asymptotically correct relative frequencies, rather than successive initial segments of one ordered sequence (Smilansky 2020).
